Dirichlet series
| L(s) = 1 | + 8·5-s − 8·7-s − 6·9-s − 12·11-s + 20·13-s + 4·17-s − 4·19-s − 8·23-s + 18·25-s + 4·27-s + 8·29-s + 4·31-s − 64·35-s + 8·37-s − 12·41-s + 4·43-s − 48·45-s − 20·47-s + 36·49-s + 40·53-s − 96·55-s − 4·59-s − 8·61-s + 48·63-s + 160·65-s − 28·67-s + 16·71-s + ⋯ |
| L(s) = 1 | + 3.57·5-s − 3.02·7-s − 2·9-s − 3.61·11-s + 5.54·13-s + 0.970·17-s − 0.917·19-s − 1.66·23-s + 18/5·25-s + 0.769·27-s + 1.48·29-s + 0.718·31-s − 10.8·35-s + 1.31·37-s − 1.87·41-s + 0.609·43-s − 7.15·45-s − 2.91·47-s + 36/7·49-s + 5.49·53-s − 12.9·55-s − 0.520·59-s − 1.02·61-s + 6.04·63-s + 19.8·65-s − 3.42·67-s + 1.89·71-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{80} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{80} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{80} \cdot 7^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.15186\times 10^{14}\) |
| Root analytic conductor: | \(7.56549\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{80} \cdot 7^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(5.409418268\) |
| \(L(\frac12)\) | \(\approx\) | \(5.409418268\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 7 | \( ( 1 + T )^{8} \) | |
| good | 3 | \( 1 + 2 p T^{2} - 4 T^{3} + 22 T^{4} - 4 p^{2} T^{5} + 58 T^{6} - 176 T^{7} + 130 T^{8} - 176 p T^{9} + 58 p^{2} T^{10} - 4 p^{5} T^{11} + 22 p^{4} T^{12} - 4 p^{5} T^{13} + 2 p^{7} T^{14} + p^{8} T^{16} \) |
| 5 | \( 1 - 8 T + 46 T^{2} - 196 T^{3} + 718 T^{4} - 2276 T^{5} + 6522 T^{6} - 16768 T^{7} + 39442 T^{8} - 16768 p T^{9} + 6522 p^{2} T^{10} - 2276 p^{3} T^{11} + 718 p^{4} T^{12} - 196 p^{5} T^{13} + 46 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 11 | \( 1 + 12 T + 120 T^{2} + 76 p T^{3} + 5008 T^{4} + 25012 T^{5} + 110776 T^{6} + 431548 T^{7} + 1515358 T^{8} + 431548 p T^{9} + 110776 p^{2} T^{10} + 25012 p^{3} T^{11} + 5008 p^{4} T^{12} + 76 p^{6} T^{13} + 120 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} \) | |
| 13 | \( 1 - 20 T + 242 T^{2} - 2104 T^{3} + 14614 T^{4} - 84480 T^{5} + 420214 T^{6} - 140340 p T^{7} + 6996354 T^{8} - 140340 p^{2} T^{9} + 420214 p^{2} T^{10} - 84480 p^{3} T^{11} + 14614 p^{4} T^{12} - 2104 p^{5} T^{13} + 242 p^{6} T^{14} - 20 p^{7} T^{15} + p^{8} T^{16} \) | |
| 17 | \( 1 - 4 T + 36 T^{2} - 12 p T^{3} + 1420 T^{4} - 6148 T^{5} + 35260 T^{6} - 140460 T^{7} + 681382 T^{8} - 140460 p T^{9} + 35260 p^{2} T^{10} - 6148 p^{3} T^{11} + 1420 p^{4} T^{12} - 12 p^{6} T^{13} + 36 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 19 | \( 1 + 4 T + 106 T^{2} + 392 T^{3} + 5390 T^{4} + 17608 T^{5} + 173462 T^{6} + 487988 T^{7} + 3893458 T^{8} + 487988 p T^{9} + 173462 p^{2} T^{10} + 17608 p^{3} T^{11} + 5390 p^{4} T^{12} + 392 p^{5} T^{13} + 106 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 23 | \( 1 + 8 T + 100 T^{2} + 600 T^{3} + 4648 T^{4} + 25432 T^{5} + 158988 T^{6} + 793096 T^{7} + 4198990 T^{8} + 793096 p T^{9} + 158988 p^{2} T^{10} + 25432 p^{3} T^{11} + 4648 p^{4} T^{12} + 600 p^{5} T^{13} + 100 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 29 | \( 1 - 8 T + 136 T^{2} - 744 T^{3} + 8404 T^{4} - 35416 T^{5} + 338520 T^{6} - 1205752 T^{7} + 10809670 T^{8} - 1205752 p T^{9} + 338520 p^{2} T^{10} - 35416 p^{3} T^{11} + 8404 p^{4} T^{12} - 744 p^{5} T^{13} + 136 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 31 | \( 1 - 4 T + 116 T^{2} - 292 T^{3} + 7660 T^{4} - 556 p T^{5} + 363628 T^{6} - 667188 T^{7} + 12635238 T^{8} - 667188 p T^{9} + 363628 p^{2} T^{10} - 556 p^{4} T^{11} + 7660 p^{4} T^{12} - 292 p^{5} T^{13} + 116 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 37 | \( 1 - 8 T + 128 T^{2} - 968 T^{3} + 10276 T^{4} - 66296 T^{5} + 581248 T^{6} - 3372408 T^{7} + 24790566 T^{8} - 3372408 p T^{9} + 581248 p^{2} T^{10} - 66296 p^{3} T^{11} + 10276 p^{4} T^{12} - 968 p^{5} T^{13} + 128 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 41 | \( 1 + 12 T + 252 T^{2} + 2020 T^{3} + 23932 T^{4} + 134252 T^{5} + 1230436 T^{6} + 5322212 T^{7} + 49511686 T^{8} + 5322212 p T^{9} + 1230436 p^{2} T^{10} + 134252 p^{3} T^{11} + 23932 p^{4} T^{12} + 2020 p^{5} T^{13} + 252 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} \) | |
| 43 | \( 1 - 4 T + 224 T^{2} - 668 T^{3} + 23824 T^{4} - 52140 T^{5} + 1619632 T^{6} - 2724756 T^{7} + 80014686 T^{8} - 2724756 p T^{9} + 1619632 p^{2} T^{10} - 52140 p^{3} T^{11} + 23824 p^{4} T^{12} - 668 p^{5} T^{13} + 224 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 47 | \( 1 + 20 T + 444 T^{2} + 5796 T^{3} + 76348 T^{4} + 746420 T^{5} + 7220260 T^{6} + 55738788 T^{7} + 423816262 T^{8} + 55738788 p T^{9} + 7220260 p^{2} T^{10} + 746420 p^{3} T^{11} + 76348 p^{4} T^{12} + 5796 p^{5} T^{13} + 444 p^{6} T^{14} + 20 p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( 1 - 40 T + 944 T^{2} - 15976 T^{3} + 215620 T^{4} - 2431704 T^{5} + 23723408 T^{6} - 204071384 T^{7} + 1568992486 T^{8} - 204071384 p T^{9} + 23723408 p^{2} T^{10} - 2431704 p^{3} T^{11} + 215620 p^{4} T^{12} - 15976 p^{5} T^{13} + 944 p^{6} T^{14} - 40 p^{7} T^{15} + p^{8} T^{16} \) | |
| 59 | \( 1 + 4 T + 266 T^{2} + 688 T^{3} + 34798 T^{4} + 54096 T^{5} + 3056918 T^{6} + 3110036 T^{7} + 203887378 T^{8} + 3110036 p T^{9} + 3056918 p^{2} T^{10} + 54096 p^{3} T^{11} + 34798 p^{4} T^{12} + 688 p^{5} T^{13} + 266 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( 1 + 8 T + 6 p T^{2} + 2292 T^{3} + 60526 T^{4} + 303652 T^{5} + 6159546 T^{6} + 25551600 T^{7} + 439390098 T^{8} + 25551600 p T^{9} + 6159546 p^{2} T^{10} + 303652 p^{3} T^{11} + 60526 p^{4} T^{12} + 2292 p^{5} T^{13} + 6 p^{7} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 67 | \( 1 + 28 T + 552 T^{2} + 7860 T^{3} + 104224 T^{4} + 1197188 T^{5} + 12702408 T^{6} + 116708268 T^{7} + 1007083326 T^{8} + 116708268 p T^{9} + 12702408 p^{2} T^{10} + 1197188 p^{3} T^{11} + 104224 p^{4} T^{12} + 7860 p^{5} T^{13} + 552 p^{6} T^{14} + 28 p^{7} T^{15} + p^{8} T^{16} \) | |
| 71 | \( 1 - 16 T + 552 T^{2} - 6864 T^{3} + 131788 T^{4} - 1322128 T^{5} + 18146008 T^{6} - 148647120 T^{7} + 1592348710 T^{8} - 148647120 p T^{9} + 18146008 p^{2} T^{10} - 1322128 p^{3} T^{11} + 131788 p^{4} T^{12} - 6864 p^{5} T^{13} + 552 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} \) | |
| 73 | \( 1 - 16 T + 416 T^{2} - 5296 T^{3} + 86284 T^{4} - 901648 T^{5} + 11082784 T^{6} - 97127472 T^{7} + 972956070 T^{8} - 97127472 p T^{9} + 11082784 p^{2} T^{10} - 901648 p^{3} T^{11} + 86284 p^{4} T^{12} - 5296 p^{5} T^{13} + 416 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 + 344 T^{2} + 704 T^{3} + 63740 T^{4} + 170560 T^{5} + 8074984 T^{6} + 22406016 T^{7} + 736732678 T^{8} + 22406016 p T^{9} + 8074984 p^{2} T^{10} + 170560 p^{3} T^{11} + 63740 p^{4} T^{12} + 704 p^{5} T^{13} + 344 p^{6} T^{14} + p^{8} T^{16} \) | |
| 83 | \( 1 - 8 T + 246 T^{2} - 12 p T^{3} + 25750 T^{4} - 40580 T^{5} + 1876234 T^{6} + 2703720 T^{7} + 123032194 T^{8} + 2703720 p T^{9} + 1876234 p^{2} T^{10} - 40580 p^{3} T^{11} + 25750 p^{4} T^{12} - 12 p^{6} T^{13} + 246 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 89 | \( ( 1 - 8 T + 340 T^{2} - 2104 T^{3} + 44694 T^{4} - 2104 p T^{5} + 340 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 97 | \( 1 + 36 T + 892 T^{2} + 16188 T^{3} + 246108 T^{4} + 3208884 T^{5} + 37592036 T^{6} + 406022892 T^{7} + 4110173318 T^{8} + 406022892 p T^{9} + 37592036 p^{2} T^{10} + 3208884 p^{3} T^{11} + 246108 p^{4} T^{12} + 16188 p^{5} T^{13} + 892 p^{6} T^{14} + 36 p^{7} T^{15} + p^{8} T^{16} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.23893605564431246777002267244, −2.84038199226174039143953540644, −2.82399680702691540820203196860, −2.82055047932791009824823328171, −2.80676059784463624112731466813, −2.73314180898798973132153181458, −2.70620620956903716992963900722, −2.62126998771708787225951953892, −2.43144961686806233728600646938, −2.07813094145600871741811587390, −2.00120461277695245819386701374, −1.95766369362148526571275927264, −1.91712421895055386759872029072, −1.87969639710193875641024781970, −1.84746133885732787312514480866, −1.58891309744411527482002644914, −1.27068693734392346129915572566, −1.22363493568752974706144836010, −1.11896822187715331983444412202, −0.896130684206562812326210149659, −0.819921208461384156660412432832, −0.60365327892934537888042406345, −0.52655069349625856972413198448, −0.20195567754837795987658225791, −0.19879043470868471409665375588, 0.19879043470868471409665375588, 0.20195567754837795987658225791, 0.52655069349625856972413198448, 0.60365327892934537888042406345, 0.819921208461384156660412432832, 0.896130684206562812326210149659, 1.11896822187715331983444412202, 1.22363493568752974706144836010, 1.27068693734392346129915572566, 1.58891309744411527482002644914, 1.84746133885732787312514480866, 1.87969639710193875641024781970, 1.91712421895055386759872029072, 1.95766369362148526571275927264, 2.00120461277695245819386701374, 2.07813094145600871741811587390, 2.43144961686806233728600646938, 2.62126998771708787225951953892, 2.70620620956903716992963900722, 2.73314180898798973132153181458, 2.80676059784463624112731466813, 2.82055047932791009824823328171, 2.82399680702691540820203196860, 2.84038199226174039143953540644, 3.23893605564431246777002267244
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.